Multiple Scattering in GEANT4 L aszl o Urb an, Central - - PowerPoint PPT Presentation

Multiple Scattering in GEANT4 1

Multiple Scattering in GEANT4

L´ aszl´

• Urb´

an, Central Res.Inst.Phys., Budapest 03 July 2001

Abstract MSC model in G4 : its main features, development of the model and some G4/data and G4/G3 comparisons.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Transport of charged particles: A charged particle starts from a given point ( origin of the reference frame) moving in a given direction ( dir. of the z-axis). Let p(r, d, t) denote the probability density of finding the particle at the point r = (x, y, z) moving in the direction of the unit vector d after having travelled a path length t. The problem to be solved : p(r, d, t) = ? if the initial energy of the particle, the material parameters, all the cross sections are known ...

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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The transport is governed by the transport equation ∂p(r, d, t) ∂t +∇p(r, d, t) = N

• [p(r, d

′, t)−p(r, d, t)]dσ(χ)

dΩ dΩ (1) which can be solved exactly for special cases only, but this equation can be used to derive different moments of p. The practical solutions of the particle transport can be classified as

• detailed (microscopic) simulations,
• condensed simulations
• and mixed simulation algorithms.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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detailed simulation : exact, but time consuming if the energy is not small condensed simulation: simulates the global effects of the collisions during a step, but uses approximations mixed algorithms: ”hard collisions” are simulated one by one + global effects of the ”soft collisions”. ⇒ Detailed simulation is used for low energy particles only ⇒ examples of the condensed simulations : EGS,GEANT3 - both use Moliere theory, GEANT4 ⇒ mixed simulation algorithm is used e.g in PENELOPE.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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MSC model in GEANT4 : Notations: true path length or ’t’ path length is the total length travelled by the particle. All the physical processes restrict this ’t’ step. geometrical or ’z’ path length is the straight distance between the starting and endpoint of the step, if there is no magnetic

• field. The geometry gives a constraint for this ’z’ step.

path length correction(transformation): t ⇐ ⇒ z t = ⇒ z : F(z, t) z = ⇒ t : G(t, z) scattering angle distribution: f(x, t), x = cosθ lateral displacement : R(r, t).

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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PHYSICS INPUT: (first)transport mean free path 1/λ = 2πN 1

−1

(1 − cosχ)dσ(χ) dΩ d(cosχ) (2) where dσ(χ)

dΩ

is the differential cross section of the scattering, N = NAvogadro

ρ A ,

ρ is the density of the material, A is the atomic weight, NAvogadro is the Avogadro’s number . i-th transport mean free path is defined similarly with the substitution (1 − cosχ) = ⇒ (1 − Pi(cosχ)), Pi(cosχ) - i-th Legendre polynomial. Instead of using the cross section directly the model uses λ and λ2 to calculate the different (spatial and angle) distributions.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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steps of MSC algorithm ( are essentially the same for many condensed simulation):

• 1. selection of step length ⇐

= physics processes + geometry (MSC performs the t ⇐ ⇒ z transformations only)

• 2. transport to the initial direction

(not MSC business)

• 3. sample scattering angle θ
• 4. compute lateral displacement, relocate particle

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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STEP 1

• 1. take the smallest step length coming from the step

limitations given by the physics processes (all but MSC) t = min(tproc1, tproc2, ..., tprocn)

• 2. do the t → z transformation zphys ⇐

= F(z, t)

• 3. ask step limit zgeom from geometry
• 4. take the final (geom.) step size as zstep = min(zphys, zgeom)
• 5. compute the corresponding true step length

tstep ⇐ = G(t, zstep)

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Model functions F(z, t) and G(t, z):

• 1. G4 standard (until now): not distributions, mean values
• nly

z = F(z, t) = λ ∗ (1 − exp(−t/λ)) t = G(t, z) = −λ ∗ ln(1 − z

λ)

( the formulae come from the theory).

• 2. G4 new : distributions with the theoretical mean values

F(z, t) : F(u) = β2 ∗ u ∗ exp(−β ∗ u) , where u = t

z − 1

(0 ≤ u < ∞) , β is computed from < u >=

t <z> − 1

G(t, z) : G(v) = γ2 ∗ v ∗ exp(−γ ∗ v) , where v = t

z − 1

(0 ≤ v < ∞) , γ is computed from < v >= <t>

z

− 1

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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STEP 3 sample scattering angle θ from the model distribution f(x, t) (x = cosθ). f(x, t) :

• 1. G4 standard (until now):

f(x, t) = p ∗ (a + 1)2 ∗ (a − 1)2 2 ∗ a ∗ 1 (a − x)3 + (1 − p) ∗ 1 2 (3) where a = 1 + α ∗ t

λ, α = 0.9, 0 ≤ p ≤ 1 .

• 2. G4 new :

f(x, t) = q∗f0(x, t)+(1−q)∗{p∗f1(x, t)+(1−p)∗f2(x, t)} (4) where 0 ≤ p, q ≤ 1, fi(x, t)-s are relatively simple functions

• f x and the variable τ = t

λ.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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STEP 4 compute the mean lateral displacement according to the theoretical formula and change the position of the particle correspondingly. note: this step is executed only if the particle is ’far’ from the boundary of the volume.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Angle distributions, G4 new

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Angle distributions, G4 new

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Energy deposit, G4 new,G3 and data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Energy deposit, G4 new,G3 and data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Transmission, G4 new,G3 and data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Backscattering is a difficult problem for condensed simulations. One step to the good direction in the new G4 MSC algoritm: limit the step in MSC when entering a volume . (This is NOT the user limit, the step is limited by MSC near to boundaries

• nly!) Algorithm:

tlim = max(λ, tmin) where tmin = 0.001 micrometer. if( safety < tlim) and (tstep > tlim) tstep = fact ∗ λ and MSC limits the step. This means a restriction of the step length for low energy particles only. Some results of this very simple algorithm follow ...

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Backscattering, low energy e-/e+

backscattering coeff. of e-/e+ backscattered from gold

10 20 30 40 50 60 1 10 10 2 E(keV) cback %

G3 G4new G4std data

10 20 30 40 50 60 1 10 10 2 E(keV) cback %

G3 G4new G4std data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Backscattering, not so low energy

backscattering coeff. of e- backscattered from Al

2 4 6 8 10 12 14 10 -1 1 10 E(MeV) cback %

G3 G4new G4std data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Backscattering, Z dependence

backscattering coeff. of 1 MeV e- for diff. materials

5 10 15 20 25 30 35 40 45 50 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90Z cback %

G3 G4new data

5 10 15 20 25 30 35 40 45 50 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90Z Eback %

G3 G4new data

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Backscattering, energy spectra

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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The new MSC version brought -sometime big- changes in the physics results for low energy particles. Is there any effect/change for a high energy setup ? The answer is yes. Example : 30(23 mm Fe + 0.4 mm Si) SICAPO calorimeter, 6 GeV e- initiated showers.

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Showers in sampling calorimeters 1000 showers/run with 0.01 mm cut.(diff en.cut in Fe/Si in G3!) program version Evis in MeV RMS (MeV) ex.time(sec) GEANT3 32.14 3.78 970. G4 std 31.71 3.62 1520. G4 new 32.42 3.89 1600. G4 new(no bound) 32.00 3.79 1600. = ⇒

• more Evis and bigger RMS in G4 new than in G4 std
• G4 new is slower than G4 std by 5 % and this change in

speed is not due to the boundary algorithm

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Stability of the results, cut dependence

Cut dependence of Evis/RMS for 6 GeV e- showers

28 29 30 31 32 33 34 10 -2 10 -1 1 10 mm Evis(MeV)

G3 G4new G4std

3.4 3.6 3.8 4 4.2 4.4 4.6 10 -2 10 -1 1 10 mm RMS(MeV)

G3 G4new G4std

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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→ cut dependence of Evis seen in ATLAS HEC and EM barrel for electrons is probably cured with the new MSC !!!! another exercise: 20 GeV muons in the same calorimeter, 10000 showers/run (motivation: EvisG3 > EvisG4 by 3 % in the ATLAS TileCal Fe + Scint calorimeter) results (for cut = 1 mm) program Evis(MeV) RMS(MeV) ∆ = 100 ∗ Evis−EvisG3

EvisG3

G3 6.17 3.21

• G4 std

6.03 2.93

• 2.3

G4 new 6.18 3.36 +0.2 → G4 std gives smaller Evis than G3 while EvisG4new = EvisG3 ( stat.error of Evis ∼ 0.5 % here.) The results are similar for cut = 0.1 mm).

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Conclusions:

• the new MSC version exists and works rather well
• G4 with the new MSC gives better physics results than G4

std ( and at least for the simple setups simulated here G4 new results are better than G3 results)

• G4 new is sligthly slower then G4 with standard MSC
• backscattering can be simulated

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Things to do:

• write-up (preprint,paper) - top priority
• results =

⇒ EM Web gallery

• some more tuning on angle distribution (tail is
• verestimated, but G3 tends to underestimate the tail ...)
• correlations (e.g. z - cosθ)
• ...

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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How to use the new MSC version: it is in the CVS HEAD under the name G4MultipleScatteringx

• update your working directory
• perform a change G4MultipleScattering =

⇒ G4MultipleScatteringx in your PhysicsList Try to use this new MSC, I need your feedback and sooner or later this will be the default !

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Angle distributions with G4 std MSC

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Angle distributions with G4 std MSC, motivation for MSC development!

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001

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Angle distributions with G3

Geant4 Workshop 2001, Genova L.Urb´ an(CRIP,Budapest) 03 July 2001